# 代数内部

${\displaystyle \operatorname {core} (A):=\left\{x_{0}\in A:\forall x\in X,\exists t_{x}>0,\forall t\in [0,t_{x}],x_{0}+tx\in A\right\}}$[4]

## 性质

${\displaystyle A,B\subset X}$则：

• ${\displaystyle A}$吸收的当且仅当${\displaystyle 0\in \operatorname {core} (A)}$ [1]
• ${\displaystyle A+\operatorname {core} B\subset \operatorname {core} (A+B)}$[5]
• ${\displaystyle A+\operatorname {core} B=\operatorname {core} (A+B)}$如果${\displaystyle B=\operatorname {core} B}$[5]

### 和内部的关系

${\displaystyle X}$拓扑向量空间${\displaystyle \operatorname {int} }$表示内部算子，且${\displaystyle A\subset X}$，则有：

• ${\displaystyle \operatorname {int} A\subseteq \operatorname {core} A}$
• 如果${\displaystyle A}$是非空凸集且${\displaystyle X}$ 有限维的，则有${\displaystyle \operatorname {int} A=\operatorname {core} A}$[2]
• 如果${\displaystyle A}$是有非空内部的凸集，则有${\displaystyle \operatorname {int} A=\operatorname {core} A}$[6]
• 如果${\displaystyle A}$是闭凸集且${\displaystyle X}$完备度量空间，则有${\displaystyle \operatorname {int} A=\operatorname {core} A}$[7]

## 参考文献

1. Jaschke, Stefan; Kuchler, Uwe. Coherent Risk Measures, Valuation Bounds, and (${\displaystyle \mu ,\rho }$)-Portfolio Optimization. 2000.
2. Aliprantis, C.D.; Border, K.C. Infinite Dimensional Analysis: A Hitchhiker's Guide 3rd. Springer. 2007: 199–200. ISBN 978-3-540-32696-0. doi:10.1007/3-540-29587-9.
3. ^ John Cook. Separation of Convex Sets in Linear Topological Spaces (pdf). May 21, 1988 [November 14, 2012]. （原始内容存档 (PDF)于2019-02-27）.
4. ^ Nikolaĭ Kapitonovich Nikolʹskiĭ. Functional analysis I: linear functional analysis. Springer. 1992. ISBN 978-3-540-50584-6.
5. Zălinescu, C. Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing Co., Inc. 2002: 2–3. ISBN 981-238-067-1. MR 1921556.
6. ^ Shmuel Kantorovitz. Introduction to Modern Analysis. Oxford University Press. 2003: 134. ISBN 9780198526568.
7. ^ Bonnans, J. Frederic; Shapiro, Alexander, Perturbation Analysis of Optimization Problems, Springer series in operations research, Springer, Remark 2.73, p. 56, 2000 [2016-12-19], ISBN 9780387987057, （原始内容存档于2019-05-02）.